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Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2\sqrt{2}}}\Big)$
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Answer

Let $\text{u}=\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$
Put $2\text{x}=\cos\theta$
$\Rightarrow\text{u}=\sin^{-1}\Big(2\times\cos\theta\sqrt{1-\cos^2\theta}\Big)$
$\Rightarrow\text{u}=\sin^{-1}(2\cos\theta\sin\theta)$
$\Rightarrow\text{u}=\sin^{-1}(\sin2\theta)\ .....(\text{i})$
Let, $\text{v}=\sqrt{1-4\text{x}^2}\ .....(\text{ii})$
Here,
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}}\Big)$
$\Rightarrow2\text{x}\in\Big(-\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}\Big)$
$\Rightarrow\theta\in\Big(\frac{\pi}{4},\frac{3\pi}{4}\Big)$
So, from equation (i),
$\text{u}=\pi=2\theta$
$\Big[\text{Since},\sin^{-1}(\sin\theta)=\pi-\theta,\text{ if }\theta\in\Big(\frac{\pi}{2},\pi\Big)\Big]$
$\Rightarrow\text{u}=\pi-2\cos^{-1}(2\text{x})\big[\text{Since},2\text{x}=\cos\theta\big]$
Differentiating it with respect to x,
$\frac{\text{du}}{\text{dx}}=0-2\bigg(\frac{-1}{\sqrt{1-(2\text{x})^2}}\bigg)\frac{\text{d}}{\text{dx}}(2\text{x})$
$\Rightarrow\frac{\text{du}}{\text{dx}}=\frac{2}{\sqrt{1-4\text{x}^2}}(2)$
$\Rightarrow\frac{\text{du}}{\text{dx}}=\frac{4}{\sqrt{1-4\text{x}^2}}\ .....(\text{iii})$
From equation (ii)
$\Rightarrow\frac{\text{dv}}{\text{dx}}=\frac{-4\text{x}}{\sqrt{1-4\text{x}^2}}$
But, $\text{x}\in\Big(-\frac{1}{2},\frac{1}{2\sqrt{2}}\Big)$
$\Rightarrow\frac{\text{dv}}{\text{dx}}=\frac{-4(-\text{x})}{\sqrt{1-4(-\text{x})^2}}$
$\Rightarrow\frac{\text{dv}}{\text{dx}}=\frac{4\text{x}}{\sqrt{1-4\text{x}^2}}\ .....(\text{iv})$
Diferentiating equation (ii) with respect to x,
$\frac{\text{dv}}{\text{dx}}=\frac{1}{2\sqrt{1-4\text{x}^2}}\frac{\text{d}}{\text{dx}}(1-4\text{x}^2)$
$\Rightarrow\frac{\text{dv}}{\text{dx}}=\frac{1}{2\sqrt{1-4\text{x}^2}}(-8\text{x})$
$\Rightarrow\frac{\text{dv}}{\text{dx}}=\frac{-4\text{x}}{\sqrt{1-4\text{x}^2}}\ .....(\text{v})$
Dividing equation (iii) by (v)
$\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}=\frac{4}{\sqrt{1-4\text{x}^2}}\times\frac{\sqrt{1-4\text{x}^2}}{-4\text{x}}$
$\therefore\frac{\text{du}}{\text{dv}}=-\frac{1}{\text{x}}$

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